A new mathematical formulation of robot and obstacles is presented such that for on-line collision recognition only robot joint positions in the workspace are required. This reduces calculation time essentially because joint positions in workspace can be computed every time from the joint variables through robot geometry. It is assumed that the obstacles in the workspace of the manipulator are represented by convex polygons. For every link of the redundant robot and every obstacle a boundary ellipse is defined in workspace such that there is no collision if the robot joints are outside this ellipsis.

In addition to this, a collision avoidance method is presented which allows the use of redundant degrees of freedom such that a manipulator can avoid obstacles while tracking the desired end-effector trajectory. The method is based on the generalized inverse with boundary ellipse functions as optimization criteria. The method permits the tip of the hand to approach any arbitrary point in the free space while the kinematic control algorithm maximizes the boundary ellipse function of the critical link. The effectiveness of the proposed methods is discussed by theoretical considerations and illustrated by simulations of the motion of three- and four-link planar manipulators between obstacles.